{"id":"https://openalex.org/W1484825648","doi":"https://doi.org/10.2307/2272162","title":"On an extension of Hilbert's second \u03b5-theorem","display_name":"On an extension of Hilbert's second \u03b5-theorem","publication_year":1975,"publication_date":"1975-09-01","ids":{"openalex":"https://openalex.org/W1484825648","doi":"https://doi.org/10.2307/2272162","mag":"1484825648"},"language":"en","primary_location":{"id":"doi:10.2307/2272162","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272162","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109842705","display_name":"T. B. Flannagan","orcid":null},"institutions":[{"id":"https://openalex.org/I124357947","display_name":"University of London","ror":"https://ror.org/04cw6st05","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947"]},{"id":"https://openalex.org/I4210160447","display_name":"Universidad de Londres","ror":"https://ror.org/054v4yq51","country_code":"MX","type":"education","lineage":["https://openalex.org/I4210160447"]},{"id":"https://openalex.org/I98640878","display_name":"Bedford College","ror":"https://ror.org/01aa0nz04","country_code":"GB","type":"education","lineage":["https://openalex.org/I4405252469","https://openalex.org/I98640878"]}],"countries":["GB","MX"],"is_corresponding":true,"raw_author_name":"T. B. Flannagan","raw_affiliation_strings":["Bedford College, University of London, London, England"],"affiliations":[{"raw_affiliation_string":"Bedford College, University of London, London, England","institution_ids":["https://openalex.org/I98640878","https://openalex.org/I124357947","https://openalex.org/I4210160447"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5109842705"],"corresponding_institution_ids":["https://openalex.org/I124357947","https://openalex.org/I4210160447","https://openalex.org/I98640878"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.05914176,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"40","issue":"3","first_page":"393","last_page":"397"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9958999752998352,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6756299734115601},{"id":"https://openalex.org/keywords/ackermann-function","display_name":"Ackermann function","score":0.6221643686294556},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.5916541218757629},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5359348654747009},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.49231278896331787},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4722670316696167},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4238743782043457},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.41230806708335876},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3677406907081604},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.09972947835922241},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.076510488986969},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.05970624089241028}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6756299734115601},{"id":"https://openalex.org/C33436860","wikidata":"https://www.wikidata.org/wiki/Q341835","display_name":"Ackermann function","level":3,"score":0.6221643686294556},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.5916541218757629},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5359348654747009},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.49231278896331787},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4722670316696167},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4238743782043457},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.41230806708335876},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3677406907081604},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.09972947835922241},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.076510488986969},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.05970624089241028},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2272162","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272162","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1480405493","https://openalex.org/W1536240750","https://openalex.org/W1975810242","https://openalex.org/W2587962973","https://openalex.org/W4231621694"],"related_works":["https://openalex.org/W2076743759","https://openalex.org/W4235144492","https://openalex.org/W4388435768","https://openalex.org/W2756792282","https://openalex.org/W2756744637","https://openalex.org/W2759493398","https://openalex.org/W3111457560","https://openalex.org/W3112295066","https://openalex.org/W4210343663","https://openalex.org/W2373903136"],"abstract_inverted_index":{"The":[0,63,83],"aim":[1],"of":[2,15,19,30,45,60,69,76,91,112,131,155,164,179,192,220,231],"this":[3],"paper":[4],"is":[5,95,102,130,140,159,172,188,200,208,210,226],"to":[6,149],"correct":[7],"an":[8,162],"error":[9,84],"in":[10,13,73,85,88,224],"the":[11,31,55,89,103,132,150,165,173,177,190,218,229],"proof":[12,90,230],"[2]":[14,86,225],"a":[16,141,212],"strengthened":[17],"version":[18,65],"Hilbert's":[20,23],"second":[21],"\u03b5-theorem.":[22],"original":[24],"theorem":[25],"says":[26,71],"(effectively)":[27],"that":[28],"formulae":[29],"form":[32,133],"\u2203":[33,113,182],"xA":[34,38],"\u2192":[35],"A":[36,100,107,139,157,171,206],"(\u03b5":[37,117,125],")":[39,119,127,187],"can":[40,79],"be":[41,81],"eliminated":[42],"from":[43,57,106],"deductions":[44],"\u2208-free":[46],"sentences":[47],"B":[48,116,129,199],"(i.e.":[49],"those":[50],"which":[51,209],"do":[52],"not":[53,160,211],"contain":[54],"\u03b5-symbol)":[56],"collections":[58],"X":[59],"\u03b5-free":[61],"sentences.":[62],"extended":[64],"(Theorem":[66],"III.":[67],"11":[68],"[2])":[70],"that,":[72],"addition,":[74],"instances":[75],"Ackermann's":[77],"schema":[78],"also":[80],"eliminated.":[82],"occurs":[87],"Theorem":[92,232],"III.7":[93],"and":[94,137,197],"described":[96],"as":[97,222],"follows.":[98],"If":[99],"\u2032":[101,158,207],"formula":[104],"obtained":[105],"by":[108,115,123],"replacing":[109],"every":[110],"occurrence":[111],"yB":[114,118,126],"(or":[120],"\u2200":[121],"yC":[122],"C":[124,135],"if":[128,138,170],"\u00ac":[134],"),":[136,204],"Q":[142,145,174,213],"3":[143],"or":[144],"4-axiom":[146],"then,":[147],"contrary":[148],"claim":[151],"on":[152],"p.":[153,195],"73":[154],"[2],":[156],"necessarily":[161],"axiom":[163],"same":[166],"form.":[167],"For":[168],"example,":[169],"3-axiom":[175],"where":[176,198],"rank":[178,191,221],"\u03b5":[180],"x":[181],"yD":[183],"(":[184,202],"x,":[185],"y":[186],"\u2264":[189],"\u03b5yB":[193],"(see":[194],"70)":[196],"D":[201],"t,y":[203],"then":[205],"3-axiom.":[214],"In":[215],"other":[216],"words,":[217],"notion":[219],"defined":[223],"inadequate":[227],"for":[228],"III.11.":[233]},"counts_by_year":[{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}